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Chrome File Edit View History Bookmarks People Window Help * 4) 56% , Tue 9:56 PM a @ E Pedro Secure https://piazza resources s3.amazonaws.com/j6wopzgl2h2ujjaloabxlko16ic/e3.pdf?AWSAccessKeyld-AKIAIE DNRLJ4AZKBW6HA&Expires-151253.; D e3.pdf 417 Problem 4. Recall that is an ordered basis of Ps. Another ordered basis for P is Let T: B B be the linear transformation defined by T(p(p() 2p(r). If it's not obvious to you that this is linear, check it. A. Find the matrix of T with respect to the basis P, i.e, find Tpp. B. Find the matrix of T with respect to the basis 2, e, find Too C. Let g(z) be the element ofPg with1g(z)le=[-2 1 3]T. Find [T(g(z))le Write g(x) and T(g(E)) as linecbinaions of O

Explanation / Answer

A.If p(x) = 1, then p’(x) = 0 so that T(p(x))= p’(x)+2p(x) = 0+2=2.

If p(x) = x, then p’(x) = 1 so that T(p(x))= p’(x)+2p(x) =1+2x.

If p(x) = x2, then p’(x) = 2x so that T(p(x))= p’(x)+2p(x) = 2x+2x2.

Then [T]PP =

2

1

0

0

2

2

0

0

2

B. If p(x) =2x2-x+3, then p’(x) = 4x-1 so that T(p(x))= p’(x)+2p(x) =4x-1+2(2x2-x+3) = 5+4x2.

If p(x) =x2-1, then p’(x) = 2x so that T(p(x))= p’(x)+2p(x) =2x+2(x2-1) = -2+2x+2x2.

If p(x) =3x2-2x+2, then p’(x) = 6x-2 so that T(p(x))= p’(x)+2p(x) =6x-2+2(3x2-2x+2) = 2+2x+6x2.

Then [T]QQ =

5

-2

2

0

2

2

4

2

6

C. If [g(x)]Q = (-2,1,3)T, then g(x) = -2(2x2-x+3)+1( x2-1)+3(3x2-2x+2) = 6x2-4x-1. Then g’(x) = 12x-4 so that T(g(x))= g’(x)+2g(x) =12x-4 +2(6x2-4x-1) = 12x2+4x-6.

Now, let M =

2

1

3

6

12

-1

0

-2

-4

4

3

-1

2

-1

-6

The RREF of M is

1

0

0

-2

32/5

0

1

0

1

74/5

0

0

1

3

-26/5

Now, g(x)=-2(2x2-x+3)+1( x2-1)+3(3x2-2x+2) and [T(g(x)]=(32/5)( 2x2-x+3)+(74/5)( x2-1)-(26/5)( 3x2-2x+2) i.e. [T(g(x)]Q = (32/5,74/5,-26/5)T.

2

1

0

0

2

2

0

0

2

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